The generalized recurrent set, explosions and Lyapunov functions

Abstract : We consider explosions in the generalized recurrent set for homeomorphisms on a compact metric space. We provide multiple examples to show that such explosions can occur, in contrast to the case for the chain recurrent set. We give sufficient conditions to avoid explosions and discuss their necessity. Moreover, we explain the relations between explosions and cycles for the generalized recurrent set. In particular, for a compact topological manifold with dimension greater or equal 2, we characterize explosion phenomena in terms of existence of cycles. We apply our results to give sufficient conditions for stability, under C 0 perturbations, of the property of admitting a continuous Lyapunov function which is not a first integral.
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https://hal-univ-avignon.archives-ouvertes.fr/hal-02097706
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Soumis le : vendredi 12 avril 2019 - 10:55:35
Dernière modification le : mardi 16 avril 2019 - 01:27:51

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  • HAL Id : hal-02097706, version 1

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Olga Bernardi, Anna Florio, Jim Wiseman. The generalized recurrent set, explosions and Lyapunov functions. 2019. ⟨hal-02097706⟩

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